INTRODUCTION

In this chapter we extend the instrumental variables analyses discussed in Chapter 23 to allow for two-sided noncompliance in a randomized experiment. In the discussion on one-sided noncompliance, only those units assigned to the active treatment could choose whether or not to comply with their assignment. Now we allow for the possibility that some of the units assigned to the control group do in fact receive the active treatment. In terms of the notation introduced in Chapter 23, we allow the value of the potential receipt of treatment given assignment to the control group, Wi(0), to be 1. This generalization implies that there are now possibly four different compliance types, defined by the pair of values of potential treatment responses, (Wi(0), Wi(1)), instead of two as in the onesided compliance case. As in Chapter 23, these compliance types play a key role in our analysis.

Critical again in our analysis are assumptions about the absence of effects of assignment on the primary outcome for subgroups for which the assignment has no effect on the receipt of treatment. These are assumptions that we referred to as exclusion restrictions in the previous chapter. A new type of assumption in this chapter is what we refer to as monotonicity. This assumption rules out the presence of units who always, in this experiment, that is, under both values of the assignment, do the opposite of their assignment; such units are characterized by Wi(z) = 1 − z for z = 0, 1, that is, Wi(0) = 1 and Wi(1) = 0. Units with such compliance behavior are sometimes referred to as defiers. The monotonicity assumption, which rules out the presence of these defiers, implies that Wi(z) is weakly monotone in z for all units and is also referred to as the no-defier assumption. In many applications this assumption is a pausible one, but in some cases it can be controversial. In the previous chapter it was satisfied by construction because no one assigned to the control group could receive the active treatment. In the two-sided noncompliance setting, monotonicity is a substantive assumption that need not always be satisfied.